How does a single neuron in hidden layer affect training accuracy
The claim that Neural Network with a single hidden layer can model any functions is proven in Cybenko's Approximation by superpositions of a sigmoidal function.
https://link.springer.com/article/10.1007/BF02551274
check also: https://en.wikipedia.org/wiki/Universal_approximation_theorem
The thing is that the neural network using sigmoidal functions, which are non-linear functions can.